On Infinite Words Determined by Indexed Languages

Smith, Tim

doi:10.1007/978-3-662-44522-8_43

Cited by 5 publications

(7 citation statements)

References 16 publications

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“…1 We require that a nonterminal can only be replaced by a terminal word if it has no index attached to it. It is easy to see that this leads to the same languages [33]. Derivation trees are always unranked trees with labels in N I * ∪ T ∪ {ε} and a very straightforward analog to those of context-free grammars.…”

Section: Indexed Languagesmentioning

confidence: 94%

“…In the case L ⊆ a * , the SUP is just the finiteness problem, for which Hayashi presented a procedure using his pumping lemma [18]. However, neither Hayashi's nor any of the other pumping or shrinking lemmas [12,24,29,33] appears to yield decidability of the SUP. Therefore, this work employs a different approach: Given an indexed grammar G with L(G) ⊆ a * 1 · · · a * n , we apply a series of transformations, each preserving the simultaneous unboundedness (sections 3.3 to 3.7).…”

Section: Indexed Languagesmentioning

confidence: 99%

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An Approach to Computing Downward Closures

Zetzsche

2015

Lecture Notes in Computer Science

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Abstract. The downward closure of a word language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of any language is regular. While the downward closure appears to be a powerful abstraction, algorithms for computing a finite automaton for the downward closure of a given language have been established only for few language classes.This work presents a simple general method for computing downward closures. For language classes that are closed under rational transductions, it is shown that the computation of downward closures can be reduced to checking a certain unboundedness property.This result is used to prove that downward closures are computable for (i) every language class with effectively semilinear Parikh images that are closed under rational transductions, (ii) matrix languages, and (iii) indexed languages (equivalently, languages accepted by higher-order pushdown automata of order 2).

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Section: Indexed Languagesmentioning

confidence: 94%

Section: Indexed Languagesmentioning

confidence: 99%

An Approach to Computing Downward Closures

Zetzsche

2015

Lecture Notes in Computer Science

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“…It is hoped that work in this area will help to build up a theory of the complexity of infinite words with respect to what language classes can determine them. See [17] and [18] for progress along these lines.…”

Section: Resultsmentioning

confidence: 99%

“…In Smith [17], prefix languages are used to characterize the infinite words determined by several classes of one-way stack automata, and also studied in connection with multihead deterministic finite automata. In Smith [18], prefix languages are used to characterize the infinite words determined by the indexed languages of Alfred Aho.…”

Section: Related Workmentioning

confidence: 99%

On infinite words determined by L systems

Smith

2015

Theoretical Computer Science

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A deterministic L system generates an infinite word α if each word in its derivation sequence is a prefix of the next, yielding α as a limit. We generalize this notion to arbitrary L systems via the concept of prefix languages. A prefix language is a language L such that for all x, y ∈ L, x is a prefix of y or y is a prefix of x. Every infinite prefix language determines a single infinite word. Where C is a class of L systems (e.g. 0L, DT0L), we denote by ω(C) the class of infinite words determined by the prefix languages in C . This allows us to speak of infinite 0L words, infinite DT0L words, etc. We categorize the infinite words determined by a variety of L systems, showing that the whole hierarchy collapses to just three distinct classes of infinite words: ω(PD0L), ω(D0L), and ω(CD0L). Our results are obtained with the help of a pumping lemma which we prove for T0L systems.

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“…where * ⇒ denotes the reflexive transitive closure of ⇒. Derivation trees are always unranked trees with labels in 𝑁 𝐼 * ∪ 𝑇 ∪ {𝜀} and a very straightforward analog to those of context-free grammars (see, for example, [72]). If 𝑡 is a labeled tree, then its yield, denoted yield(𝑡), is the word spelled by the labels of its leaves.…”

Section: Basic Notions: Slices and Indexed Grammarsmentioning

confidence: 99%

Separability by piecewise testable languages and downward closures beyond subwords

Zetzsche

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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Indexed languages are a classical notion in formal language theory. As the language equivalent of second-order pushdown automata, they have received considerable attention in higher-order model checking. Unfortunately, counting properties are notoriously difficult to decide for indexed languages: So far, all results about non-regular counting properties show undecidability.In this paper, we initiate the study of slice closures of (Parikh images of) indexed languages. A slice is a set of vectors of natural numbers such that membership of 𝒖, 𝒖 +𝒗, 𝒖 +𝒘 implies membership of 𝒖 + 𝒗 + 𝒘. Our main result is that given an indexed language 𝐿, one can compute a semilinear representation of the smallest slice containing 𝐿's Parikh image.We present two applications. First, one can compute the set of all affine relations satisfied by the Parikh image of an indexed language. In particular, this answers affirmatively a question by Kobayashi: Is it decidable whether in a given indexed language, every word has the same number of 𝑎's as 𝑏's.As a second application, we show decidability of (systems of) word equations with rational constraints and a class of counting constraints: These allow us to look for solutions where a counting function (defined by an automaton) is not zero. For example, one can decide whether a word equation with rational constraints has a solution where the number of occurrences of 𝑎 differs between variables 𝑋 and 𝑌 .

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On Infinite Words Determined by Indexed Languages

Cited by 5 publications

References 16 publications

An Approach to Computing Downward Closures

An Approach to Computing Downward Closures

On infinite words determined by L systems

Separability by piecewise testable languages and downward closures beyond subwords

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